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More generally, given a subgroup $\\Lambda$ of $\\Gamma$, the pair $(C^*_r(\\Gamma),C^*_r(\\Lambda))$ has Property (T) if and only if the pair of groups $(\\Gamma, \\Lambda)$ has Property (T)."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0505189","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.OA","submitted_at":"2005-05-10T17:35:10Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"207ade6879a4df32052eb649a6228502403447e008682a935ba47264c0800d14","abstract_canon_sha256":"837cee0f1b5191239b5e35dd874fb32703c7e9953b1bf7e6f4612caf77eb28a2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:59.030925Z","signature_b64":"Gdhgz2nssV17cuPB+RO++fRvQL/WJPpAmr7HEMn2xY/8Du5gN+GVH2J9SBNL3hR8wWQYtLfldtd9+XGX4M4iBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"230fc7648e4b8870b8f304458ba42ec9341244d828f203cb2057be2420d9c66c","last_reissued_at":"2026-05-18T02:27:59.030359Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:59.030359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A property (T) for C*-algebras","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"Bachir Bekka","submitted_at":"2005-05-10T17:35:10Z","abstract_excerpt":"We define a notion of Property (T) for an arbitrary $C^*$-algebra $A$ admitting a tracial state. We extend this to a notion of Property (T) for the pair $(A,B),$ where $B$ is a $C^*$-subalgebra of $A.$ Let $\\Gamma$ be a discrete group and $C^*_r(\\Gamma)$ its reduced algebra. We show that $C^*_r(\\Gamma)$ has Property (T) if and only if the group $\\Gamma$ has Property (T) . More generally, given a subgroup $\\Lambda$ of $\\Gamma$, the pair $(C^*_r(\\Gamma),C^*_r(\\Lambda))$ has Property (T) if and only if the pair of groups $(\\Gamma, \\Lambda)$ has Property (T)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0505189","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0505189","created_at":"2026-05-18T02:27:59.030437+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0505189v1","created_at":"2026-05-18T02:27:59.030437+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0505189","created_at":"2026-05-18T02:27:59.030437+00:00"},{"alias_kind":"pith_short_12","alias_value":"EMH4OZEOJOEH","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_16","alias_value":"EMH4OZEOJOEHBOHT","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_8","alias_value":"EMH4OZEO","created_at":"2026-05-18T12:25:53.335082+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EMH4OZEOJOEHBOHTARCYXJBOZE","json":"https://pith.science/pith/EMH4OZEOJOEHBOHTARCYXJBOZE.json","graph_json":"https://pith.science/api/pith-number/EMH4OZEOJOEHBOHTARCYXJBOZE/graph.json","events_json":"https://pith.science/api/pith-number/EMH4OZEOJOEHBOHTARCYXJBOZE/events.json","paper":"https://pith.science/paper/EMH4OZEO"},"agent_actions":{"view_html":"https://pith.science/pith/EMH4OZEOJOEHBOHTARCYXJBOZE","download_json":"https://pith.science/pith/EMH4OZEOJOEHBOHTARCYXJBOZE.json","view_paper":"https://pith.science/paper/EMH4OZEO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0505189&json=true","fetch_graph":"https://pith.science/api/pith-number/EMH4OZEOJOEHBOHTARCYXJBOZE/graph.json","fetch_events":"https://pith.science/api/pith-number/EMH4OZEOJOEHBOHTARCYXJBOZE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EMH4OZEOJOEHBOHTARCYXJBOZE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EMH4OZEOJOEHBOHTARCYXJBOZE/action/storage_attestation","attest_author":"https://pith.science/pith/EMH4OZEOJOEHBOHTARCYXJBOZE/action/author_attestation","sign_citation":"https://pith.science/pith/EMH4OZEOJOEHBOHTARCYXJBOZE/action/citation_signature","submit_replication":"https://pith.science/pith/EMH4OZEOJOEHBOHTARCYXJBOZE/action/replication_record"}},"created_at":"2026-05-18T02:27:59.030437+00:00","updated_at":"2026-05-18T02:27:59.030437+00:00"}